Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.util.Scanner;

/**
 * Created by Harry on 4/2/20.
 */
//https://atcoder.jp/contests/dp/tasks/dp_i
public class Main {
    public static void main(String[] args){
        Scanner scanner = new Scanner(System.in);
        int N = scanner.nextInt();
        double[] p = new double[N];
        for(int i=0; i<N; i++){
            p[i] = scanner.nextDouble();
        }
        double[] dp = new double[N+1];
        dp[0] = 1;
        for(int i=0; i<N; i++){
            for(int j=i+1; j>=0; j--){
                dp[j] = dp[j]*(1-p[i])+(j-1>=0 ? dp[j-1]*p[i] : 0);
            }
        }
        double res = 0.0;
        for(int i=(N+1)/2; i<=N; i++){
            res += dp[i];
        }
        System.out.print(res);
    }
}